Scalar Product and Angle Between Vectors
Use the dot product to connect vectors, angles and perpendicularity.
Formula
\[\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta\]
For components,
\[\begin{pmatrix}a_1\a_2\a_3\end{pmatrix}\cdot\begin{pmatrix}b_1\b_2\b_3\end{pmatrix}=a_1b_1+a_2b_2+a_3b_3.\]
If \(\mathbf a\cdot\mathbf b=0\), the vectors are perpendicular.
Worked example
Let \(\mathbf a=\begin{pmatrix}1\2\2\end{pmatrix}\) and \(\mathbf b=\begin{pmatrix}2\0\1\end{pmatrix}\).
\[\mathbf a\cdot\mathbf b=1(2)+2(0)+2(1)=4.\]
Also \(|\mathbf a|=3\) and \(|\mathbf b|=\sqrt5\), so
\[\cos\theta=\frac{4}{3\sqrt5}.\]